Skew Boolean Algebras
A zero element in a skew lattice S is an element 0 of S such that for all, or, dually, . (0)
A Boolean skew lattice is a symmetric, distributive normal skew lattice with 0, such that is a Boolean lattice for each . Given such skew lattice S, a difference operator \ is defined on by x\ y = where the latter is evaluated in the Boolean lattice . In the presence of (D3) and (0), \ is characterized by the identities: and One thus has a variety of skew Boolean algebras characterized by identities (D3), (0) and (S B). A primitive skew Boolean algebra consists of 0 and a single non-0 D-class. Thus it is the result of adjoining a 0 to a rectangular skew lattice D via (0) with, if and otherwise. Every skew Boolean algebra is a subdirect product of primitive algebras. Skew Boolean algebras play an important role in the study of discriminator varieties and other generalizations in universal algebra of Boolean behavior. For more details on skew Boolean algebras see .
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